Sum of Interior Angles of Polygons
2013-02-14 06:58
This GeoGebra applet dynamically demonstrates the theorem for the sum of interior angles of polygons. By constructing arbitrary simple or convex polygons, it automatically calculates and displays the measure of each interior angle and their sum. Users can drag vertices to alter the polygon's shape and observe the mathematical relationship between the sum of angles and the number of sides, namely (
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<input a0="H" a1="G" a2="F"/>
<output a0="a7"/>
</command>
<element type="angle" label="a7">
<angleStyle val="0"/>
<value val="0.7175340673596463"/>
<show object="true" label="false"/>
<condition showObject="box1 ∧ (n > 7)"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-35" y="5"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="I" a1="H" a2="G"/>
<output a0="a8"/>
</command>
<element type="angle" label="a8">
<angleStyle val="0"/>
<value val="0.26499491781256684"/>
<show object="true" label="false"/>
<condition showObject="box1 ∧ (n > 8)"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-30" y="4"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="J" a1="I" a2="H"/>
<output a0="a9"/>
</command>
<element type="angle" label="a9">
<angleStyle val="0"/>
<value val="0"/>
<show object="false" label="false"/>
<condition showObject="box1 ∧ (n > 9)"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-28" y="10"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="B" a1="A" a2="Element[listP, n]"/>
<output a0="a1"/>
</command>
<element type="angle" label="a1">
<angleStyle val="0"/>
<value val="1.884955592153876"/>
<show object="true" label="false"/>
<condition showObject="box1"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-52" y="22"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="A" a1="Element[listP, n]" a2="Element[listP, n - 1]"/>
<output a0="aLast"/>
</command>
<element type="angle" label="aLast">
<angleStyle val="0"/>
<value val="1.884955592153876"/>
<show object="false" label="false"/>
<condition showObject="box1"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-35" y="-3"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="TableText">
<input a0="{Join[{First[{a1, a2, a3, a4, a5, a6, a7, a8, a9}, n - 1], {aLast}}]}" a1=""vc""/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<condition showObject="box1 ∧ box2 ∧ simple"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="0"/>
<absoluteScreenLocation x="815" y="10"/>
</element>
<command name="PointIn">
<input a0="poly1"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="false" ev="4"/>
<condition showObject="box3"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="5.551115123125783e-17" y="-1.1102230246251565e-16" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[P, Element[{A, B, C, D, E, F, G, H, I, J}, i]]" a1="i" a2="1" a3="n"/>
<output a0="listPartition"/>
</command>
<element type="list" label="listPartition">
<show object="true" label="true" ev="4"/>
<condition showObject="box3"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="3"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="boolean" label="box3">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="box1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelOffset x="590" y="65"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[P,RandomPointIn[poly1]]"/>
<caption val="Dissect into triangles"/>
</element>
<command name="Circle">
<input a0="(0, 0)" a1="1"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Sequence">
<input a0="Segment[listP(i), listP(i + 1)]" a1="i" a2="1" a3="n"/>
<output a0="listSeg"/>
</command>
<element type="list" label="listSeg">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="button" label="button0">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelOffset x="869" y="9"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[lang,If[lang==1,2,1]] RunClickScript[buttonLang]"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<caption val="中文"/>
</element>
<element type="button" label="buttonLang">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelOffset x="869" y="70"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetCaption[button0,If[lang==1,"ENG","中文"]] SetCaption[listSide,If[lang==1,"邊的數目=","Number of sides="]] SetCaption[box1,If[lang==1,"顯示內角","Show interior angles"]] SetCaption[box3,If[lang==1,"分割成三角形","Dissect into triangles"]] SetCaption[box2,If[lang==1,"顯示內角數值","Show angle measures"]] "/>
<font serif="false" sizeM="2" size="20" style="0"/>
</element>
<element type="numeric" label="lang">
<value val="2"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="6"/>
<labelMode val="2"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="If">
<input a0="simple" a1="If[convex, If[lang ≟ 1, "這是凸多邊形。", "This is a convex polygon."], If[lang ≟ 1, "這是凹多邊形。", "This is a concave polygon."]]" a2="If[lang ≟ 1, "這不是簡單多邊形。", "This is not a simple polygon."]"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="6"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<absoluteScreenLocation x="18" y="98"/>
</element>
</construction>
</geogebra>